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Considerable attention has been paid to dynamic searchable symmetric encryption (DSSE) which allows users to search on dynamically updated encrypted databases. To improve the performance of real-world applications, recent non-interactive multi-client DSSE schemes are targeted at avoiding per-query interaction between data owners and data users. However, existing non-interactive multi-client DSSE schemes do not consider forward privacy or backward privacy, making them exposed to leakage abuse attacks. Besides, most existing DSSE schemes with forward and backward privacy rely on keeping a keyword operation counter or an inverted index, resulting in a heavy storage burden on the data owner side. To address these issues, we propose a non-interactive multi-client DSSE scheme with small client storage, and our proposed scheme can provide both forward privacy and backward privacy. Specifically, we first design a lightweight storage chain structure that binds all keywords to a single state to reduce the storage cost. Then, we present a Hidden Key technique, which preserves non-interactive forward privacy through time range queries, ensuring that data with newer timestamps cannot match earlier time ranges. We conduct extensive experiments to validate our methods, which demonstrate computational efficiency. Moreover, security analysis proves the privacy-preserving property of our methods.